Question

triangle abc lies on the xy plane. the coordinates of a point of the line segment bc are (-2, -3). if the coordinates of point a are (-7, -6) and the midpoint of ab is (-5, -9), what are the coordinates of point c?

Explanation:

Step1: Define midpoint formula

For midpoint $M(x_m,y_m)$ of points $(x_1,y_1)$ and $(x_2,y_2)$:
$$x_m=\frac{x_1+x_2}{2}, \quad y_m=\frac{y_1+y_2}{2}$$

Step2: Assign known values to formula

Let $B=(-2,3)$, $C=(x,y)$, midpoint $A=(-7,-6)$. Substitute into $x$-formula:
$$-7=\frac{-2+x}{2}$$
Substitute into $y$-formula:
$$-6=\frac{3+y}{2}$$

Step3: Solve for $x$-coordinate of $C$

Multiply both sides by 2:
$$-14=-2+x$$
Rearrange to solve for $x$:
$$x=-14+2=-12$$

Step4: Solve for $y$-coordinate of $C$

Multiply both sides by 2:
$$-12=3+y$$
Rearrange to solve for $y$:
$$y=-12-3=-15$$

Answer:

$(-12, -15)$